Instead of learning a discrete-time representation it is also possible to learn an ODE. In general, this is less straightforward because the data available tends to be the state at different points in time. However, if the time derivative is available Gonzalez Garcia et al. [1]} showed that a NN with states and derivative inputs could learn the ODE.
A similar, but more interpretable approach is “Sparse Identification of Nonlinear Dynamics" (SINDy)[2]}. This approach involves selecting a dictionary of candidate functions for Eq. REF , and using sparse regression to identify the dominant terms which most accurately reconstruct the time derivative. Alternatively, the time derivative can be approximated with a multistep time-integration scheme and a NN can be trained off of this approximation [3]}. A drawback of all these approaches is the data needs to be close in time to get good approximations of the derivative, if the time derivative is not known.
